Thanks, that was very useful! I do have a little better understanding of those lift and drag coefficients now, so the adventure continues...Broomstick wrote:I found this link, not sure how helpful you'll find it.
Let us consider a simple model, where a fixed-wing aircraft is in a state of level flight (i.e. engine thrust vector is perfectly orthogonal to the vector of gravity) in perfectly still air (no wind) at constant airspeed (no acceleration/deceleration along thrust axis). There would be five forces acting upon the aircraft – W (weight, directed toward the center of gravity), B and L (buoyant force and dynamic lift, both directed opposite to the vector of W), Θ (thrust produced by aircraft engines through power input, orthogonal to the vector of W by definition of “level flight”) and D (drag, the aerodynamic force opposite to the vector of Θ). The aircraft would be in a state of aerodynamic equilibrium when net effect of all those forces is zero, i.e.:
W = B + L and D = Θ
Since magnitude of B would usually be negligible unless the atmosphere is extremely dense (e.g. 97% CO₂ at 92 ATA pressure, like near the surface of Venus) we shall not consider the effect of buoyant force for the moment and reduce the first equation to W = L. Since:
W = m0·g
where m0 is total mass of the aircraft and g is the gravity acceleration, and:
L = ρ·v²·Aw·CL/2
where ρ is the density of air, v is airspeed, Aw is planform area (more or less wing area, for most airplane designs) and CL is the lift coefficient (which turns out to be mainly a function angle of attack – AOA, wing aspect ratio – AR and airfoil profile – AFP). Greater AR and AOA as well as more efficient AFP shapes will increase lift, but there is a limit of maximum AOA value known as “stall point” (beyond which, lift will be drastically lost). Maximum value of AR will be constrained by resistance of available materials to structural stress and other practical design considerations. CL is also inversely related to Mach number (Ma), which may be computed as following:
Ma = v/a
where a is the speed of sound in a given environment. Let us recall that the speed of sound under given atmospheric conditions is directly related to air temperature (T) and, usually, more or less inversely proportional to the mean mole mass of the air mixture (M). However, since effect of Ma is negligible as long as the speed of any local airflows does not exceed a, we shall not discuss it for now and consider that CL is purely a function of the airframe shape and AOA in subsonic regime. Thus we can calculate the lifting efficiency of the aircraft as:
m0 = ρ/g·v²·Aw·CL(AOA+,AR+,AFP)/2
Thus, in theory, lifting efficiency of the same aircraft design at the same airspeed depends solely on ρ/g ratio and overall performance should be unaffected if ρ and g are increased or decreased by the same magnitude, while any increase of ρ greater then increase of g and any decrease of g greater than decrease of ρ should improve it.
Now, let us consider drag, which basically consists of two separate components: Dp – pressure drag (resulting from the oncoming airflow) and Df – skin friction. Thus:
D = Dp + Df
Pressure drag is usually the most significant component of D in subsonic regime and may be computed as following:
Dp = ρ·v²·Ac·CDp/2
where Ac is the frontal profile cross-section area (which can be approximated to the area of 2D projection of the aircraft when looking at it from the front). Note that greater AR and AOA of the wings would lead to increase of Ac and, therefore, directly contribute to the pressure drag. CDp is the pressure drag coefficient – mainly a function of the airframe shape.
Second component of the drag force is the skin friction, which is usually negligible at lower airspeeds but may become a factor as the speed of oncoming airflow increases.
Df = ρ·v²·As·CDf/2
CDf has an inverse relation to Reynolds number (Re) which may computed as following:
Re = ρ·v·λ/μ
where μ is the dynamic viscosity of air at applicable temperature/pressure conditions and λ is length of the object (usually mean chord length for thin airfoils and nose-to-tail length for the fuselage). Since values of λ would vary significantly for different parts of the airframe, overall Re value would have to be computed as weighted average of Re for each of those parts based on their skin surface ratio. Unfortunately, as I have mentioned earlier, it is impossible to accurately predict the value of μ without conducting lab experiments, unless the atmosphere is composed of nearly pure substance and pressure does not exceed about 3 ATA.
Note, that As has a direct relation to Aw, while the value of λ for wings has an inverse relation to AR (higher AR will result in smaller chord lengths). Thus:
D = Dp + Df = ρ·v²·(Ac(AOA+,AR+)·CDp + As(Aw) ·CDf(μ+,ρ-,v-,λ(AR-)-)/2
As we can see, most factors that affect lift, also affect drag in the same way, even though airframes are usually designed in such a manner that impact of AOA and AR on drag will be of much lesser magnitude then on lift. While overall impact of airspeed (v) on drag is partially offset by its inverse relation to CDf, the magnitude of this offset is relatively negligible. The impact of ρ on CDf is not obvious, since all factors that affect the value of ρ (i.e. air composition, pressure and temperature) also affect the value of μ, but the latter impact is not easily predictable except for a very limited subset of possible atmospheric conditions. However, since overall impact of CDf on the value of D is relatively minor at subsonic airspeeds, we may assume that the net effect of any changes in values of ρ and μ on CDf would be negligible. Thus, for subsonic regimes, both CDp and CDf may be considered merely functions of the airframe shape and we could reduce the above to:
D = Dp + Df ≈ ρ·v²·(Ac(AOA+,AR+)·CDp + As(Aw+)·CDf(AR+)/2 = ρ·v²·XD(AOA+,AR+)
where XD(Aw+,AOA+,AR+) is a function of the airframe geometry that has direct relations to Aw, AOA and AR, but of significantly lesser magnitude then XL(Aw+++,AOA++,AR++) = Aw·CL(AOA+,AR+)/2. However note, while lift must counteract weight, and aircraft lifting efficiency is directly proportional to ρ/g ratio, thrust-drag relation is unaffected by the value of g. Moreover, while D is directly proportional to ρ, it has no immediately obvious impact on the value of Θ, which depends on the efficiency of the powerplants and the particular propulsion method used. Even though some factors that affect ρ might have certain effect on those parameters as well, the magnitude of such impact would, in any case, be considerably less significant.
Till now, we assumed that the vector of lift would be exactly opposite to the vector of weight and that vectors of thrust and drag are completely orthogonal to the weight-lift axis. However, this is not usually the case in practice as the aircraft would normally require a non-zero AOA in order to achieve useful lift – which would require the chord line of the main lifting palnform to be tilted slightly backward in respect to the axis of thrust. Such a tilt, in its turn, produces wingtip vortices, which somewhat deflect the airflow downward from the normal “free stream” vector which would be parallel to the vector of thrust. Since the effective lifting force produced by the planform is orthogonal to the vector of actual airflow, it would be tilted backward by the same angle in relation to the nominal weight-lift axis. As a consequence, only a part of the vector of effective lift (Le) would be acting in the nominal direction of lift (L), while the difference would be added to the nominal vector of drag (D). This difference is called lift-induced drag (Di). Thus:
L = Le - Di and D = De + Di
where Le is the effective lift produced by the planform and De is the effective drag produced as the reaction to the oncoming airflow (Dp + Df, often known as “parasite drag”). By the same token, the actual vector of drag produced by the tilted planform would tilt downward by the same angle and, thus, should be subtracted from the nominal vector of drag and added to the nominal vector of weight (hence creating “drag-induced weight” – Wi). However, airfoils are normally designed to maximize lift and to minimize drag, so the latter effect can usually be negligible.
To complicate the matters further, non-zero AOA is usually produced by pitching the aircraft nose upward, which means that actual vector of thrust will be tilted from the nominal thrust-drag axis by the angle approximately equal to AOA. Thus, a part of the thrust force must be subtracted from the nominal vector of thrust and added to the nominal vector of lift (thrust-induced lift – Li). But, since vector of weight is always directed toward the center of gravity regardless of the aircraft orientation, there will be no such thing as weight-induced thrust. Thus we obtain the following aerodynamic equilibrium equations:
L = Le + Li - Di = W = We + Wi
Θ = Θe - Li = D = De + Di - Wi
Since Wi would usually be negligible for most practical purposes our drag equation may thus be expressed as following:
D = ρ·v²·XD(Aw+,AOA+,AR+) + ρ·v²·(2·δ/π)·XL(Aw+++,AOA++,AR++) = ρ·v²·ZD(Aw+,AOA+,AR+)
where δ is the angle (in radians) by which oncoming airflow of the planform deviates from the nominal axis of thrust-drag (≈ AOA/2) and ZD = XD + (2·δ/π)·XL (which is chiefly a function of planform area, angle of attack, aspect ratio and other parameters of the airframe geometry). Thus, the amount of power input required to overcome drag under specific airspeed and atmospheric conditions would be:
PD = v·D = ρ·v³·ZD(Aw+,AOA+,AR+)
While we have seen the impact of various factors on lifting performance and drag on a dynamic fixed-wing aircraft, there is another important aspect to consider – the structural stress. The airframe is subjected to various forces (i.e. weight, lift, drag, thrust) during the flight and it must be strong enough to resist them. Theoretically, the balance of forces acting on the airframe in a state of aerodynamic equilibrium is zero – but such would only be the case if your airframe was a perfect sphere. Unfortunately, perfect spheres do not make very efficient airframes, so various forces acting on the aircraft would not be evenly distributed across the structure. If we consider a typical airliner design, effect of the lifting force would be most significant on the wings, engine pylons would be subject to most substantial impact of the thrust, while the fuselage would experience most of the weight and pressure drag. And then, each time you want to execute some kind of maneuver, you must take the aircraft out of aerodynamic equilibrium which would subject it to additional structural stress.
As we have seen, g is what affects weight, while ρ is what matters most for aerodynamic forces. While both of those factors would balance each other in terms of overall lifting performance, magnitude increase in either would result in greater differential between the opposite forces and thus, more structural stress. So, if you try to fly a Boeing 737 on a planet where both gravity and atmospheric pressure are increased exactly by factor 2 compared to Earth, the overall lifting performance shall, theoretically, remain unchanged. However, structural stress on fuselage/wing joints shall be increased approximately by factor 4 (increase of both lift and weight forces by factor 2 will have additive effect on structural stress, given uneven distribution of forces). Likewise, the drag would be increased by factor 2, so you shall need twice more thrust to maintain the same airspeed – which would mean 4-fold stress increase on engine pylons (and further increase of stress on fuselage/wing joints since the engines of 737 are attached to the wings and most of the drag comes from the fuselage). And that’s just during level flight at constant airspeed in perfectly still air – so imagine how much extra stress your fuselage/wing joints will experience during take-off with lots of side wind! So, you would likely need to reinforce the airframe (thus increase its mass and decrease the useful payload), even though the basic design would, in theory, be just as efficient. Not to mention you’d need to burn much more fuel to fly the same distance.
If we double the value of ρ while maintaining the same value of g as on Earth, that would improve lifting performance of the aircraft, but also increase the amount of drag and structural stress. But, since we have more lift, the latter effect may be easily compensated by making the airframe stronger and heavier without sacrificing useful payload. Increase of drag may also be partially compensated by lower AR of the wings and use smaller AOA under similar flight conditions. However, since majority of the drag in subsonic regime comes from the fuselage pressure drag, overall fuel efficiency would still likely suffer – longer and more slender fuselage might make sense under such conditions.
Reducing the value of ρ by half while leaving the value of g unchanged would have the opposite effect – less lift per airspeed per wing area, but less drag and structural stress (you have such conditions on Earth at about 6,500 m altitude). Most Earth designs could easily fly if drop-launched, but many would not be able to take off on their own (airliners and airlifters could likely take off unloaded, but fighters and sport planes would never leave the ground). But since you have less drag and structural stress, you could increase wing AR and make the airframes lighter.
Any increase of g without changing the value of ρ would have a directly adverse effect on heavier-then-air aircraft (you’d need more lift to fly with the same mass, all other things being equal), while any decrease of g would have a clearly positive effect. In the Solar System, Titan is the best environment for airplanes, provided you can make engines that would work in such atmosphere (one could probably use atmospheric methane as fuel while carrying oxidizer in the “fuel tanks”).
Now, let’s consider the rotorcraft (i.e. helicopters). Basic principles of lift and drag still apply, but there are important differences. Helicopter’s “wings” are its rotor blades and you generate lift by spinning the rotor. Unlike for airplanes, forward speed of the aircraft and airspeed of the lifting planform are two completely different things. Moreover, not all sections of the airfoil have the same airspeed – the closer you get to the tip of the blade, the more airflow you have. To complicate the matters further, during forward flight, airspeed of the aircraft itself must be added to that of the advancing blade and subtracted from that of the retreating blade. Again, drag of the airframe and drag of the blades must be considered separately from each other.
Speed of sound becomes a very important factor as you are running a great risk of some local airflows becoming supersonic, which is something you normally want to avoid (as that would radically change aerodynamic characteristics of the concerned section of the airfoil). Greater speed of sound would enable you to make rotor blades longer and/or spin them faster, while reduced speed of sound would put much more constrains on the blade AR and rotor RPM rates. As we have seen, the speed of sound is a function of temperature, mole mass and specific heat capacity, i.e.:
a = √(T/(M/R - M/cp)) = √(σ·T)
where T is the temperature of air, M – the mean mole mass of the air mixture, cp – the specific heat capacity under constant pressure (given in J/mol·K), R – the ideal gas constant and σ = 1/(M/R - M/cp) – a value, which is constant for any given substance or mixture and, usually, more or less inversely proportional to the mole mass for most common atmospheric gases. Hydrogen (H₂) has the highest possible value of σ = 5.8, while for sulfur hexafluoride (SF₆) this value is very low (σ = 0.06). For Earth air, σ = 0.4 which is very close to the value for nitrogen (N₂, σ = 0.42), while for Martian and Venusian atmospheres it would be much closer to that of carbon dioxide (CO₂, σ = 0.24). So, at 20 °C, Mach 1 on Earth corresponds to airspeed of 343 m/s (1,235 km/h) but this value would be around 270 m/s (972 km/h) on Mars and Venus under the same conditions – meaning that helicopter rotor blades would need to be about 20% shorter or spin rates would need to be reduced by 20% (not a problem on Venus, but quite inconvenient on Mars).
Impact of the air density on overall performance of a rotorcraft would also be somewhat different. Helicopters have much lower lift to drag ratio then airplanes, since mean airspeed of the planform section is mostly independent from the airspeed of the airframe and you only need to take the drag coefficient of the lift-generating airfoil itself (i.e. rotor blade) into account. Yes, greater density would still mean more drag from the aircraft body during forward flight, but rotorcraft usually cruise at much lower airspeeds then fixed-wing aircraft, since they do not gain any significant lift advantage from flying faster and their maximum airspeed is constrained by the dissymmetry of lift (effect of true airspeed difference between advancing and retreating blades). So, form drag is much less of an issue.
Increased structural stress from increased air density and/or gravity would have more or less the same impact, but most of this impact would be on the rotor assembly. On the other hand, increased lifting efficiency that comes with air density increase (unless you also have gravity increase of the same or greater magnitude) would let you make rotor blades shorter and stronger to compensate for the extra stress. The inverse is not true for rotorcraft, however – while airplanes can compensate for deceased air density by higher AR wings, the AR of helicopter rotor blades is constrained by the speed of sound. Multiple rotors (e.g. transverse or tandem designs) would likely be needed even for lighter helicopter models in low-density environments. Thus, efficiency of the rotorcraft (as generalized aircraft type) clearly improves with increased air density as well as with decreased mole mass of the air – high pressure helium/hydrogen atmospheres would be ideal environments for helicopters. Of course, gravity is also a very important factor, since it has approximately the same impact on efficiency of any aircraft relying primarily on dynamic lift. Mars is a very bad place for choppers, while Venus and Titan are excellent environments (though, on Venus, you’ve got no reactive compounds in the atmosphere so you would have to carry both fuel and oxidizer on board in order to make internal combustion engines work – but hey, you could afford it, you’ve got plenty of lift and relatively low g!).
Finally, let us consider airships based on the “light gas balloon” principle (since “hot air balloon” principle is significantly less efficient as it requires constant power input to maintain buoyancy, while “low pressure balloon” principle is impractical for most gaseous environments). Let us recall the aerostatic equilibrium equation for such aircraft:
m0 ≈ p·VL·(M - ML)/(R·T) = p·ΔM·VL/(R·T)
where p is atmospheric pressure (presumed to be equal to the pressure of the lifting gas), T – temperature of the outside air (presumed to be equal to the temperature of the lifting gas), VL – volume of the lifting gas (presumed to be approximately equal to the volume of displaced air), M – the mean mole mass of the air mixture, ML – mole mass of the lifting gas, R – the ideal gas constant, and ΔM = M - ML (a value, which would be more or less constant for an aerostat using any specific lifting gas in any given environment). Since none of those parameters have any direct relation to g, the state of aerostatic equilibrium (and thus the overall aerostatic performance) is theoretically unaffected by gravity. Even though air density is a function of p, T and M (ρ = p·M/(R·T)), we must consider those parameters separately – while any change of p/T ratio will have equal impact on atmospheric density and aerostatic performance of a “light gas balloon”, such would not be the case for the value of M.
From this we can see, that airships clearly benefit from higher pressure and lower temperature. They also benefit from “heavier” air (higher M), but any increase or decrease of the value of M by factor x, will only increase/decrease aerostatic efficiency of the aircraft by factor y which would always be less than x. We must also keep in mind that we cannot have a lifting gas any “lighter” than molecular hydrogen (H₂), thus no aerostat based on “light gas balloon” principle would be able to perform unless mean mole mass of the atmospheric air is substantially greater then MH₂ (≈ 2 g/mol). Jupiter and Saturn are terrible places to fly an airship and no increase of density through increase of p/T ratio would fix that. Uranus and Neptune are slightly better, but not by much – you would still need an enormous hydrogen-filled balloon to lift the mass of cat (helium would not work at all – a helium-filled balloon will have negative ΔM in 80% H₂ “air” and, therefore, negative buoyancy). Martian atmosphere is also awful environment for ballooning – even though the air is relatively “heavy” and cold, the pressure is much too low.
Even though gravity has no direct impact on aerostatic equilibrium, it does have a few implications for airships. First of all, both buoyant force (B) and weight (W) that are acting on the aerostat are directly proportional to g. The effects of gravity cancel each other out when W = B (i.e. aerostatic equilibrium), but the effective buoyant force (Be = B - W) which acts on the airship as long as the equilibrium state is not achieved is directly proportional to g as well. This means that higher gravity would cause greater acceleration during ascent and descent and more structural stress. The latter effect would still persist in equilibrium state – just as in case of an airplane weight and lift are not evenly distributed across the airframe, weight and buoyancy are not evenly distributed across the structure of an airship. Another effect of gravity would be pressure variance with altitude, which has an inverse relation to g. So, higher gravity would mean that the same aerostat would be able to climb to a greater altitude.
Finally, unless your airship is hovering in perfectly still air, it would basically be subject to the same aerodynamic forces as an airplane (though not to the same extent). Even though most airships have no wings to speak of, zeppelins and blimps can still gain quite a bit of dynamic lift from their hulls at higher airspeeds. While lift coming from controlled thrust may be desirable or not depending on particular design, the effects of dynamic lift generated by wind are generally considered adverse. Since dynamic lift is countered by weight, higher gravity would give an airship better stability and overall performance of designs relying exclusively on aerostatic buoyancy would be improved. So, we can conclude, that overall effect of gravity increase would likely be more beneficial then adverse for airships, even though there would a trade off in increased structural stress.
Since moving airships are just as much subject to aerodynamic forces as airplanes and helicopters are, we must also consider effects of increased drag that comes with increased air density. An airship has pretty large skin area so, one might think, it would be subject to considerable skin friction. In practice, however, this is not normally the case since airships tend cruise at airspeeds much lower than needed for skin friction to become a significant factor. Also, higher air density would generally mean less lifting gas for the same payload – thus, less volume and smaller skin area. Pressure drag is a factor, but again – higher air density would mean less volume of lifting gas per payload and thus smaller drag profile area. At any rate, no heavier-then-air aircraft could possibly hope to match fuel efficiency of a “light gas balloon” type airship, since the latter does not require any power input in order to counteract the effects of gravity.
Airships would become very attractive means of air transport in high-density environments (as long as the atmosphere was not composed of nearly pure hydrogen/helium) and, unlike aircraft relying on dynamic lift, they would only benefit from increased gravity. Venus is the best place for ballooning in the Solar System and the newly discovered Kepler-10c “Mega Earth” might be a very good place for airships if air pressure is sufficiently high.
